Let

${\Omega}$ be a bounded, open subset of

${\mathbb{R}}^{N}$,

$T$ a positive number and

$Q=[0,T]\times {\Omega}$. Let

$p$ be a real number, with

$1<p<\infty $, and let

$1/p+1/{p}^{\text{'}}=1$. This paper deals with the parabolic initial-boundary value problem

${u}_{t}+A\left(u\right)=\mu $ in

$Q$,

$u=0$ on

$[0,T]\times \partial {\Omega}$,

$u\left(0\right)={u}_{0}$ in

${\Omega}$, where

$A$ is a nonlinear monotone and coercive operator in divergence form which acts from the space

${L}^{p}(0,T;{W}_{0}^{1,p}\left({\Omega}\right))$ into its dual

${L}^{{p}^{\text{'}}}(0,T;{W}^{-1,{p}^{\text{'}}}\left({\Omega}\right))$. The authors first introduced the notion of capacity related to the parabolic

$p$-Laplace operator. Then they proved a decomposition theorem for measures that do not charge the sets with null capacity. Using these results they proved the following Theorem: Let

$\mu $ be a bounded real measure on

$Q$ which does not charge the subsets of

$Q$ of null capacity, and let

${u}_{0}\in {L}^{1}\left({\Omega}\right)$. Then there is a unique renormalized solution

$u$ of the problem. Moreover,

$u$ satisfies the additional regularity:

$u\in {L}^{\infty}(0,T;{L}^{1}\left({\Omega}\right))$ and

$max(-k,min(k,u))\in {L}^{p}(0,T;{W}_{0}^{1,p}\left({\Omega}\right))$ for every

$k>0$.